Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,18,22,19,23,17,30)(15,24,26,35,27,34,32,20)(16,21,28,37,29,38,31,36)(25,39,41,43,42,44,40,33);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,35)(16,25,29,42)(17,26,18,32)(21,33,38,43)(22,34,30,24)(28,40,31,41)(36,39,37,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,31)(18,28)(19,29)(20,33)(22,37)(23,38)(24,39)(26,41)(27,42)(30,36)(32,40)(34,44)(35,43);(7,12)(9,11);(6,10)(9,11)(13,17,19,18)(14,22,23,30)(15,26,27,32)(16,28,29,31)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,39,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407280);